In a large bowl, mix together the cooled mashed potatoes, green onions, bacon, and cheddar cheese.
Form into small balls, then press a cube of Monterey Jack cheese into the center of each. Roll to cover completely and place on a parchment-lined baking sheet.
Cover and chill for 30 minutes.
In a small bowl, combine panko breadcrumbs, Parmesan cheese, salt, and pepper.
Place the beaten egg in a separate bowl.
Dip each chilled potato ball into the egg, then roll in the breadcrumb mixture until fully coated.
Return to the baking sheet and repeat with remaining balls.
Heat oil in a large pan to 350ºF / 175ºC.
Fry 3–4 balls at a time for a few minutes until golden brown.
Remove and place on kitchen paper to drain any excess oil.
Serve warm with ranch or mustard dressing for dipping.